A more subtle but very common type of focusing error occurs when the curvature of the cornea is different in different meridians. It will then focus a horizontal line at a different focal length from a vertical one, a condition
called astigmatism. If, for example, it has a smaller radius of curvature in the horizontal plane than in the vertical, the far point when measured with a vertical line as test object will be closer than when a horizontal line is used. It is corrected with a cylindrical lens – in effect a section cut from a cylinder, just as a spherical lens is from a sphere: it focuses only in one meridian. Opticians test for astigmatism by means of a target like the one below, called an astigmatic fan; an astigmatic subject will see some of the lines more sharply than others, and this will tell the optician the angle at which a cylindrical lens should be placed in front of the eye to make the refractive power as nearly as possible equal in all meridians. The power of the cylindrical lens that is needed to do this, together with its meridional angle, make up the cylindrical correction that is the second part of a prescription for spectacles.

Astigmatism and incorrect refractive power are not the only faults that may be found in the eye’s optics, and – as in many man-made optical systems – the cornea and lens together produce a number of different types of optical defects in addition to refractive error.

The first of these is due to the fact that the refractive indices of the various optical media of the eye depend on the wavelength of the incident light. In general, the refractive index increases with decreasing wavelength, so that blue light is refracted more than red. This phenomenon, called dispersion, means that the focal length of a lens depends on the wavelength, with blue being shorter than red, amounting to some 2 D over the whole visible spectrum. This gives rise to defects in the resultant image, in the form of coloured fringes, called chromatic aberration. So if you look at a blue object and a red object lying side by side at the same distance from the eye, they cannot both be in focus simultaneously, and a subject who is emmetropic when his far point is measured in red light will be short-sighted if it is measured in blue: his far point will then be only a metre or so away. This forms the basis of a simple clinical test for errors of refraction, consisting of an illuminated screen divided into three portions that are red, green and white: identical test figures are superimposed on each field, and the subject is simply asked which figure he sees most clearly. A myope, whose lens is too strong anyway, sees red targets more clearly than green, and vice versa for a hypermetrope; an emmetrope will see the one on the white background best.

The importance of chromatic aberration can be seen by the fact that visual acuity is improved by some 25 per cent in monochromatic yellow light. The eye mitigates the effects of chromatic aberration in two ways: by a yellow pigment over the fovea that reduces the blue component, and by the fact that very few blue cones are found in the centre of the fovea, where the finest spatial vision is found.

The other aberration is spherical aberration. We saw earlier that to make a surface bring parallel rays to a point it needs to be roughly spherical. Although ordinary
man-made lenses are nearly all spherical, simply because they’re easier to make, a spherical lens does not in fact bring rays to a point focus: they get bent too much as you go further out, and it is the resultant blur that is spherical aberration. The shape you really need is not a sphere but an ellipsoid. For surfaces that are small in comparison with their radii of curvature the difference is slight, and spherical aberrations are often negligible. But in the case of the eye, the aperture is of the same order of magnitude as the radius of curvature of the cornea, and the result is that rays entering near the periphery of the cornea are bent too much, and form a closer focus than those entering near the centre. To some extent Nature has compensated for spherical aberration, first of all by making a cornea that is not exactly spherical but tends towards the desired ellipsoid; and second, in that the refractive index of the lens is not constant throughout, but graded from a maximum of some 1.42 at its centre to about 1.39 at the edge, thus cancelling out, to some extent, the extra bending of peripheral light rays. The degrading effects of both spherical and chromatic aberration, and of other defects due to irregularities of the refracting surfaces, get worse as the pupil or aperture of the eye increases, and this in turn depends on the state of the iris: this is discussed on p. 137, in the context of visual acuity.

There are two other problems with the eye’s optics that are not exactly errors of focus, but do degrade the quality of the retinal image. When light enters the eye it tends to get scattered, particularly by the cornea and lens but also to some extent by bouncing off the back of the retina. This effectively superimposes on the image a more or less uniform background, perceived as glare, whose illuminance is of the order of 10 per cent of the mean illuminance of the retina. This means that if we look at a target whose actual contrast is 100 per cent, the effect of this scatter is to reduce the contrast of the retinal image to something nearer 90 per cent. Because of the progressive opacity of the lens, glare gets worse as you get older; it is particularly obvious when driving at night in the face of opposing headlights. The second problem is something that is caused by the pupil, called diffraction. Whenever light passes through a restricted aperture it tends to spread out and therefore degrades the retinal image: the smaller the aperture, the worse this gets. More precisely, the width of the resultant pointspread function is of the order of λ/d radians, where λ is the wavelength, and d the aperture of the system. In practice, so long as the pupil is bigger than some 3 mm, diffraction contributes rather little in comparison with the other optical problems.

Thus the ideal size for the pupil is a compromise. An important factor is the ambient light level: under bright photopic conditions the eye can take advantage of the excess light by constricting the pupil and improving the quality of the retinal image. In scotopic conditions, however, the eye needs all the light it can get and the quality of the retinal image is of secondary importance: in any case, we shall see later that the rods are not capable of passing on accurate information about the detailed structure of the retinal image. It is important to emphasize that pupil dilatation contributes very little to the enormous changes in sensitivity that accompany dark adaptation, since it can only vary the incoming light by a factor of 16 at most, or 1.2 log units. The effect of pupil diameter on visual acuity is discussed in more detail on p. 137.

Another factor is that the control of the pupil is closely linked to accommodation: when the ciliary muscle contracts in order to focus on a near object, there is normally an associated constriction of the pupil (the near reflex), which may help to increase depth of focus (see p. 137). As these responses are usually also combined with binocular convergence movements of the two eyes, the whole pattern of response (constriction, accommodation, convergence) is also known as the triple response. Under certain clinical conditions, notably in neurosyphilis, one may find that the pupillary response to near objects remains despite loss of the response to bright lights: this condition is known as the Argyll Robertson pupil, and is an important diagnostic neurological sign. The fact that pupil dilation is also a measure of general sympathetic activity and of emotional or sexual excitement also has it uses.

The effect of optical blur is relatively straightforward. Consider for instance the simplest of all visual objects, a star. Stars are so far away that they can in effect be regarded as infinitely small point sources. But the retinal image of the star will certainly not be a point, because the optics will spread the light out into a sort of heap on the retina. This distribution of light is called the pointspread function, and its size is a useful measure of how good or bad the optics are. In the human eye, under the best possible conditions,
the pointspread function has a diameter of about 1.5 arc min (measured half-way up); the worse your optics, the bigger this becomes.

This pointspread function is of fundamental importance, since it absolutely determines what sort of patterns we can see and what we can’t. For instance, if we have two stars rather than one, each with its own pointspread, then if they are far apart, they will be seen correctly as two separate stars; if closer, eventually there will be no little dip in between them, and the brain will have no way of knowing that there are two stars and not one. For normal observers, the angle of separation for which this kind of resolution can just be performed provides a quantitative measure of visual acuity. Its value – around 30-45 arc sec – is an order of magnitude greater than the width of a black line that can just be seen. This figure, that is found to apply in many similar tasks of resolution, can be taken as a measure of the visual acuity or resolving power of the eye.

We need to pause at this point to consider exactly what is meant by ‘seeing’ something. Seeing can mean detection, or resolution, or recognition. An ornithological friend points up in the sky and says ‘Can you see the crested willow-warbler?’ If you can’t, it may be either because it’s so far away you can’t detect it, or because although you can see a formless speck you can’t resolve the pattern of its markings, or finally because – although you can perceive every aspect of it – you haven’t the least idea what a crested willow- warbler is meant to look like. In each case, there is a failure to ‘see’, but for completely different reasons.

The existence of spatial spread has implications for detection as well as resolution. Since the incident energy from the point source is spread out over a larger area, the maximum intensity at the central peak is necessarily reduced, leading to a decrease in the contrast, ΔI/I, which determines whether it will be seen against its background (see p. 128). For objects of intrinsically high contrast, such as stars seen against the void of space, this will not matter much, and subjects with poor visual acuity as measured conventionally (see below) are not as bad at seeing stars as one might expect, bearing in mind the fact that such objects subtend an almost infinitely small angle. In the dark, whether one sees a star or not is almost entirely a matter of whether a sufficient number of photons from it fall upon a rod summation pool; as the sun rises its visibility depends on whether ΔI/I exceeds the threshold contrast. Thus – rather as in the case of the skin, discussed on p. 90 – we may find that we can localize a visual object to a much higher degree than our ability to tell whether there is one object or two.

While the main effect of contrast is on detection, it also has some effect on resolution. With the two stars, in a case like that of (c), it is clear that we cannot improve resolution simply by increasing the contrast, and such a stimulus may be described as absolutely unresolvable. But in an intermediate case like (b), whether resolution is possible or not will depend on the contrast of the original object as well as on the width of the pointspread function. This interaction between resolution and contrast can best be investigated by using grating patterns as test targets. A grating is simply a regular pattern of stripes; if the stripes are uniformly black and white, it is called a square-wave grating, because a plot of intensity as a function of distance across the grating would have a square-wave profile. In the same way, sinusoidal gratings have an intensity profile that is sinusoidal: there was an example on p. 128. In each case, one can describe the grating in terms of its spatial frequency (i.e. the number of cycles per degree) and its contrast (defined as the difference in intensity between peak and mean intensity divided by the mean intensity). Thus a pattern of alternate pure black and pure white strips, each 1° across, could be described as a square-wave grating of 100 per cent contrast and spatial frequency 0.5 cycles per degree. A simple experiment is to ask a subject to view a sinusoidal grating of a particular spatial frequency, and then reduce its contrast until he reports that he can no longer see it. If we plot this threshold contrast as a function of spatial frequency, we typically obtain a curve such as the one below. Because a blurred pointspread function affects high spatial frequencies much more than low ones, the contrast required to see the grating increases sharply as its frequency is increased, until at about 40-50 cycles per degree (the cut-off frequency), the subject cannot even see a grating of 100 per cent contrast. Because of the steepness of this cut-off, a small amount of extra blur causes a large increase in the
contrast needed, and so the method provides a sensitive measure of acuity. The reason for the fall-off in contrast sensitivity at low frequencies is discussed later (p. 152).

Of course, dispensing opticians don’t bother with all that. A rough-and-ready measure of visual acuity is to make up charts of letters of standardized shape and graded in size, and see at what point the patient is unable to read them. A common chart of this kind is the Snellen chart, in which rows of letters of diminishing size are to be read: by discovering the row at which the subject finally stops. Knowing the size of letters and the subject’s viewing distance, one can estimate his minimum resolvable angle. You can see that a letter E, for instance, is a bit like a miniature grating, and you need to be able to resolve it in order to read it. Each line of text is marked with distance in feet at which you should just be able to read it. If at 6 m you can only read the line for 12 m, then your acuity can be described as 6/12; a normal person is in theory therefore 6/6. But Snellen charts are very conservative, since they are calculated not on the basis of 45 but of 60 arc sec, or one minute. So at 6 m away, with very good vision you should in fact be able to read the line marked 5, in which case your vision is described as 6/5. The idea is no doubt to make the glasses the optician has just sold you seem better than they are.

The difficulty of the Snellen chart for scientific work is that the test is only partly one of resolution. Apart from the assumption that you are literate, it is also clear that some letters are recognized more easily than others because of their overall shape; and for some purposes the Landolt C chart, used in the same way, is preferable because it provides no extraneous clues to the subject. Even this is not ideal, since it is still possible to detect the overall orientation of the C even though it is not really resolved: for this purpose, simple barred patterns are better, since when blurred it is completely impossible to guess the original pattern, which is not true either of Snellen letters, or even Landolt Cs. Another kind of chart in which all the letters are the same size but are graded in contrast can be used to test for certain kinds of defects in the visual system in which sensitivity to contrast is specifically impaired.

The tests described so far are all genuine tests of acuity in that they require that detail of some kind be resolved. Other tests, that at first sight might also appear to be acuity tests, are really tests of detection or localization, and give apparent acuities far better than 30-45 arc sec. A well-known example is vernier acuity, where a subject is required to move two lines into alignment, as for instance in the scale of vernier callipers. Here one does incredibly well, typically of the order of a few seconds of arc. But the task is not resolution but localization: even if
the retinal image is blurred, one can still estimate where the peak is quite accurately. One can show that the longer the line the better one is, showing that accuracy is also being improved by averaging information over the whole line. Another pseudo-acuity task is the detection of stars, which may subtend extremely small angles at the eye. For instance, the bright star in Orion called Betelgeuse subtends only some 1/20 arc sec. But in a sense that figure is quite irrelevant: because of the pointspread, its image still has a width of 45 arc sec, and whether you detect it or not depends simply on how whether its luminance exceeds the absolute threshold, ΔI₀.

If the receptors simply had a one-to-one connection to bipolars, and bipolars to ganglion cells, then one would not expect any deterioration to occur as the neural image is passed along. We shall see later that this is perfectly true in the centre of the fovea, where acuity is highest, but is certainly not the case further out. Here one finds many receptors pooling their information, funnelling onto one bipolar, and many bipolars funnelling onto one ganglion cell. A telling statistic is that each eye has about 130 million receptors but only about one million ganglion cells. In fact the degree of convergence in the periphery is actually even bigger than that implies, because of huge overlap of receptive fields. In the periphery, a typical alpha ganglion cell may receive input from a staggering 75 000 rods, via 5000 rod bipolars. This is good news from the point of view of trying to detect things, because it increases the difference between the size of the signal you are trying to detect, and the background noise which would otherwise obscure it: in fact it accounts for nearly
all the difference in sensitivity between rods and cones (since the difference in threshold between one isolated rod and one cone is only about a log unit). But it is not so good for acuity. Because this neural pointspread is broader the further from the fovea, acuity is very much worse in the periphery than in the centre. And most important of all, as you dark-adapt, changing from central cone vision to rod vision with their enormous pooling of information, acuity drops off dramatically. If the contrast threshold as a function of spatial frequency is measured with a fixed pupil during progressive stages of dark adaptation, over 6 log units one finds a steady decrease in the cut-off frequency, the result of changes in the neural organization of the retina. One of the adaptational responses to reduced light levels, as we shall see, is an increase in the effective size of the ganglion cells’ summation pools, so that they can catch more light. But this obviously has the effect of increasing the degree of neural blur, and hence of reducing the overall acuity.

And this brings us back to the pupil once again. The fact that as you dark-adapt, the neural blur introduced by the retina increases, and begins to dwarf the real optical blur caused by bad optics, means that the pupil can now afford to get bigger. Whereas a large pupil in bright light is a very bad thing because of the effect it has in increasing the aberrations and defects of focusing, in dim light these are not so important, so that one can enjoy the benefits of having more light for the purposes of detection. Thus there is a kind of necessary reciprocal relation between sensitivity and acuity; the better you are at one, the worse you are at the other. A big pupil provides more sensitivity but worse acuity; pooling of receptors onto ganglion cells also provides more sensitivity but less acuity. We shall see other examples of this kind of trade-off later on.

To summarize, there are many factors that contribute to visual acuity, and their relative contributions depend on the state of adaptation of the eye: they are summarized in Table 7.3. In good light, an emmetrope’s acuity is limited by diffraction, and thus about as good as could be expected from an eye of the size that we actually have. But most people are not emmetropes, so that without spectacles it is refractive error that limits their acuity.